Projectile MotionVocabulary

Term DefinitionDisplacement vector

A vector that shows change in position using the number and an angle or direction

Projectile Any object moving through the air that is only affected by gravity

trajectory The parabolic path that a projectile follows

range The horizontal distance that a projectile travels

What is a displacement vector?

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A vector that shows a change in position using a magnitude and a direction or angle.

When you throw a ball it follows a curved path. The position of the ball is described by its displacement vector.

Vector Addition

Larry walks 10 meters West and 20 meters North. Use Pythagorean’s theorem to determine Larry’s displacement.

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10 m

θ = 45o

Displacement Vector of(10 m, 45o)

Pythagorean Theorem

Given Formula Solution a = 20 m a2+b2=c2 202+102=c2

b = 10 m 400 + 100 = c2

500 = c2

c = 22.36 m

Projectile Motion

What is a projectile? An object that is moving through the air only affected by gravity.

Regardless of its path, a projectile will always follow these rules:

1. The path a projectile follows is called its trajectory.

2. The trajectory of a projectile is a parabolic path.

3. The horizontal and vertical motions of a projectile are completely independent of one another.

a. In the absence of air resistance, there is no net horizontal force on the projectile; therefore the projectile travels with a constant horizontal velocity. In the picture below the horizontal velocity is constant: 5 m/s. Notice how the vertical velocity changes.

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20 m/s

10 m/s

5 m/s

5 m/s

5 m/s

5 m/s

5 m/s

- 20 m/s

- 10 m/s

- 30 m/s

b. In the absence of air resistance, gravity is the only vertical force on the projectile; therefore the projectile travels with a uniformly accelerated vertical motion. Every second, the vertical velocity of the projectile changes by 9.8 m/s (10 m/s)

4. Horizontal and vertical motion are completely independent of each other. Therefore, the velocity of a projectile can be separated into horizontal (vx) and vertical components (vy).

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vx = velocity in the x-directionvy = velocity in the y-direction

5. For a projectile beginning and ending at the same height, the time it takes to rise to its highest point equals the time it takes to fall from the highest point back to the original position.

6. For a projectile beginning and ending at the same height, the initial speed is equal to its final speed.

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Projectile Motion

What is a displacement vector? (p. 134)A vector that shows a change in position using a magnitude and a direction or angle.

What is a projectile? (p. 137)An object that is moving through the air only affected by gravity.

What is trajectory and what type of shape does a projectile follow? (p. 137)The path that a projectile follows is called its trajectory and it is a parabolic path.

What is range? (p. 137)The horizontal distance that a projectile travels.

What does it mean that the horizontal and vertical components of a projectile’s velocity are completely independent of one another? Note the horizontal velocity of the ball in the picture below. Why doesn’t it change? Why does the vertical velocity change?Having the two numbers be independent of one another means that the vertical and horizontal information can be talked about separately from each other. The horizontal velocity does not change because there is no air resistance. The vertical velocity accelerates at a rate of 10 m/s 2 due to gravity.

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At the instant a horizontally held rifle is fired over level ground, a bullet held at the side of the rifle is released and drops to the ground. Ignoring air resistance, which bullet strikes the ground first?The two objects will strike the ground at the same time because horizontal velocity does not affect how quickly up and down the object moves.

What is the equation to calculate the vertical velocity of a projectile? (p. 138)

vy= vertical velocity g = gravitational acceleration (10 m/s2) t = time

What is the equation to calculate the horizontal distance a projectile travels? (p.139)

vx= horizontal velocity dx = horizontal distance t = time

What is the equation to calculate the vertical distance a projectile travels? (p. 139)

dy = vertical distance t = time

Complete the table below calculating the horizontal and vertical positions of a ball rolling off a cliff at 20 meters per second: (use data table on p. 139)

Time (sec) HorizontalPosition (m)

Vertical drop(m)

0 0 0

1 20 4.9

2 40 19.6

3 60 44.1

4 80 78.4

5 100 122.5

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Range of a Projectile

List two factors that affect how far a projectile will travel are: (p. 140)1. Launch Angle 2. Launch Velocity 3. Air Resistance (if included in question)

Which angle below produces the furthest range? (p. 140)45 degrees give the greatest range.

What combination of angles produce the same range? Is there a pattern? (p. 140)80º and 10º also 60º and 30º. The relationship is that the angles are complimentary which means they add up to 90º.

Check Questions1. What is the word for the horizontal distance a projectile travels?Range

2. What launch angle gives a projectile its greatest range?45º

3. Why does a projectile move in a curved path?Because the projectile moves under the influence of gravity.

4. A ball kicked off the ground at an angle of 20 degrees and a ball kicked at an angle of 70 degrees have the same range.

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Horizontal and Vertical Velocities

A projectiles horizontal velocity is constant.A projectiles vertical velocity changes 9.8 m/s.

Vertical Velocity Practice Problems1. Chris drops a water balloon off the top of the school. What is the balloons

vertical velocity if it strikes the ground 5 seconds later?Looking For Given Relationship Solution

Vertical Velocity

t = 5s Vy = gt Vy = 10(5) = 50 m/s

2. A package falls off an airplane. How fast is the package traveling if it is in the air for 12 seconds?

Looking For Given Relationship Solution

Vertical Velocity

t = 12s Vy = gt Vy = 10(12) = 120 m/s

3. A skydiver jumps out of a plane. How fast is he traveling 8 seconds later right before he pulls his parachute?

Looking For Given Relationship Solution

Vertical Velocity

t = 8s Vy = gt Vy = 10(8) = 80 m/s

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vy = gt vy = vertical velocityg = 10 m/s2

t = time

Horizontal and Vertical Distance

1. While chasing the “Road Runner,” Wile E. Coyote makes wrong turn and ends up sliding horizontally off the edge of a high cliff. He leaves the edge of the cliff while traveling 35 m/s. He hits the bottom 4 seconds later.

a. Find the horizontal distance the coyote travels.

Given Formula Solution vx = 35 m/s dx = vxt dx = (35 m/s) x (4 s) = 140 mt = 4 s

a. Find the height of the cliff.Given Formula Solution t = 4 s dy = 4.9t2 dy = (4.9) x (4 s)2 = 78.4 m

2. Repeat the above problem with 5 seconds as the time. Looking For Given Relationship Solution

Horizontal Distance

Vx = 35 m/st = 5 s

dx = vx*t dx = 175 m

Looking For Given Relationship Solution

Vertical Distance

t = 5 s dy = 4.9t2 dy = 122.5 m

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dx = vxt dx = horizontal distancevx = horizontal velocityt = time

dy = 4.9*t2dy = vertical distancet = time

Class/HomeWork1. While chasing the “Road Runner,” Wile E. Coyote makes wrong turn and ends up sliding horizontally off the edge of a high cliff. He leaves the edge of the cliff while traveling with a horizontal velocity of 50 m/s. He hits the bottom 3 seconds later.

a. Find the horizontal distance the coyote travels.Looking For Given Relationship Solution

Horizontal Distance

Vx = 50 m/st = 3 s

dx = vx*t dx = 150 m

b. Find the height of the cliff.

Looking For Given Relationship Solution

Vertical Distance

t = 5 s dy = 4.9t2 dy = 44.1 m

2. Billy throws a water balloon off the top of the school in attempt to hit Mr. Clifton. If Billy throws the balloon with a horizontal velocity of 3 m/s and the balloon is in the air for 6 seconds determine the:(a) Horizontal distance the balloon travels.

Looking For Given Relationship Solution

Horizontal Distance

Vx = 3 m/st = 6 s

dx = vx*t dx = 18 m

(b) The height of the school.Looking For Given Relationship Solution

Vertical Distance

t = 6 s dy = 4.9t2 dy = 176.4 m

3. A monkey throws a banana horizontally at 10 m/s from the top of a tree. The banana hits the ground 6 seconds later.

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a. How far from the base of the tree does it land?Looking For Given Relationship Solution

Horizontal Distance

Vx = 10 m/st = 6 s dx = vx*t dx = 60 m

b. How high is the tree?Looking For Given Relationship Solution

Vertical Distance

t = 6 s dy = 4.9t2 dy = 176.4 m

c. How fast was the banana moving vertically (vy) as it hit the ground?Looking For Given Relationship Solution

Vertical Velocity

t = 6 s Vy = gt Vy = 10(6) = 60 m/s

d. How fast horizontally was the banana moving as it hit the ground? The banana was moving horizontally at 10 m/s as it hit the ground.

4. A bowling ball rolls off a high cliff at 5 m/s. Complete the chart below that describes its motion during each second it is in the air.

Time(s)

Horizontal Velocity

(m/s)

Vertical Velocity

(m/s)

Horizontal Distance

(m)

Vertical Distance

(m)

0 5 0 0 01 5 -10 5 4.92 5 -20 10 19.63 5 -30 15 44.14 5 -40 20 78.4

5. Chris throws a water balloon horizontally off the top of the school at 8 m/s attempting to hit his friend’s car. The balloon is in the air for 4 seconds before striking the ground.

a. How far away is Chris’ teacher standing? Looking For Given Relationship Solution

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Horizontal Distance

Vx = 8 m/st = 4 s

dx = vx*t dx = 32 m

b. What is the height of the high school?Looking For Given Relationship Solution

Vertical Distance

t = 4 s dy = 4.9t2 dy = 78.4 m

c. What is the vertical velocity of the ball just before it strikes the ground?Looking For Given Relationship Solution

Vertical Velocity

t = 4 s Vy = gt Vy = 10(4) = 40 m/s

d. What is the horizontal velocity of the ball just before it strikes the ground?The balloon is moving at 8 m/s because horizontal velocity is constant.

6. A ball is projected horizontally at velocity of 10 m/s from the top of a 50 m high cliff. How high is the ball 2.5 seconds after it is released?

Looking For Given Relationship SolutionVertical Distance

t = 2.5 s dy = 4.9t2 dy = 30.625 m

50 m – 30.625 = 19.375 m

7. A ball is projected horizontally at velocity of 10 m/s from the top of a 50 m high cliff. What is the horizontal distance from the base of the cliff that the object lands?

Looking For Given Relationship SolutionTime

Horizontal Distance

dy = 50 m

dx = vx * t

dy = 4.9t2

50 = 4.9t2

50/4.9 = t2

t = 3.19 s

dx = 10*3.19

t = 3.19 s

dx = 31.9 m

Investigation 6APhysics

Research QuestionHow can you predict the distance a projectile will travel?

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Toss a ball some distance and you can imagine in your mind the arc it follows, first rising then falling to the ground. The path the ball follows is the subject of this Investigation. Projectile motion describes objects moving under the influence of gravity, including thrown balls, coasting spacecraft, and even meteors.

The range is the horizontal distance a projectile travels between launch and touch down. In this experiment you will see how the range depends on the launch angle.

Procedure Part I

1. Set up by marking a tape line on the floor. This line is where you set the front edge of the launcher.

2. One person launches while several group members stand to either side to spot where the marble first touches the ground.

3. Place a marble in the barrel and then pull the pin back and slide it into one of the five slots.

4. Use your thumb to flick the pin out of the slot and launch the marble. 5. Use only the plastic marbles, and follow all safety rules. 6. Spotting the landing point is tricky. It often takes several launches to figure out where

the marble lands.

Investigate1. What two variables most affect the range of the marble? (Hint: Both are adjusted on

the Marble Launcher)

________________________________________________________________________________________________________________________

2. If you wish to study the effect of changing one variable, what must be done with the other one?

________________________________________________________________________________________________________________________

Procedure Part II1. Choose which release slot on the barrel of the marble launcher to use for all your

range measurements. 2. Measure the range every ten degrees from 10 to 90 degrees, and also at 45 degrees.

Table 1: Range versus angle data

Launch Angle(degrees)

Range(meters)

010

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203040455060708090

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Analysis1. Make a graph with launch angle on the x-axis and range on the y-axis. Use your graph

to find the angle that launches the marble with the greatest range.

2. The Marble Launcher starts the marble more than 10 cm above the floor. Suppose the marble were launched from floor level. Do you think it would go a shorter, longer, or about the same distance? Explain why.

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3. You are challenged to launch a marble to travel a distance of 3.00 meters. At what angle will you set the launcher?

4. Referring to your answer for question 3, state another angle that would give you the same result.

The Physics of GolfPhysics

Research QuestionHow does contact angle affect trajectory and distance?

Hypothesis

Procedure Part 1. Choose only one golf club to use for all your range measurements. Select an angle of 0 degrees and hit the ball. Record how far (the range) it travels in table 1 below.2. Measure the range every ten degrees from 10 to 90 degrees, and also at 45 degrees. 3. How does the angle affect the trajectory of the golf ball?4. How does the angle affect the distance (range) a golf ball is launched?

Table 1: Range versus angle data

Launch Angle(degrees)

Range(meters)

010203040455060708090

Analysis5. Make a graph with launch angle on the x-axis and range on the y-axis. Use your graph

to find the angle that launches the golf ball with the greatest range.

6. You are challenged to launch a golf ball to travel a distance of 200 meters. At what

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angle will you hit it?

7. Referring to your answer for question 3, state another angle that would give you the same result.

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